For the braised pork: Preheat oven to 325 degrees F.
Toss the pork cubes with the salt and pepper and half of the oregano in a bowl. Place the porcini mushrooms in a small bowl just large enough to accommodate them and pour the hot water over the top; let stand for 30 minutes to rehydrate.
Meanwhile, place an enameled 6-quart Dutch oven or other heavy, oven-safe pot over medium heat on the stovetop. Add the olive oil. Once the oil is hot, add the seasoned pork cubes, working in 2 batches so as to not crowd the pot. Cook on all sides, turning once, until the pieces have begun to brown lightly. Set aside the browned pork cubes in a bowl. Add the onion, carrot and celery and cook until softened and lightly caramelized, adding olive oil if bottom of pan becomes too dry. Add the garlic and continue to cook until fragrant, 1 to 2 minutes. Add the wine to the pot to deglaze, scraping loose any browned bits. Add the porcini mushrooms and their liquid (be sure to prevent the sediment at the bottom of the bowl from pouring in), the broth, tomato sauce, bay leaf and remaining oregano and stir to incorporate. Return the pork to the pot and cover with a tight-fitting lid. Place the pot in the oven and cook for 2 1/2 hours. Remove the pork and check for tenderness. If the pork is not fork-tender, return to the oven and continue to braise for another 30 to 45 minutes. Let cool slightly and taste for seasonings.
For the cavatelli pasta: Place the flour on a smooth work surface and add 1 teaspoon salt, stirring lightly with a fork to incorporate. Form the small pile of flour into a ring with room in middle to accommodate the eggs. Dump the eggs and yolks into the middle of the ring. Stir the eggs slowly with a fork to mix, then continuing with the fork, draw the flour into the eggs a little at a time, working the pile in a circular motion to bring the dough together without breaking the ring. Once the pasta dough has started to form, knead the dough, rolling and turning the ball until a smooth and silky dough is achieved. Wrap the pasta dough tightly in plastic wrap and let rest for at least 30 minutes or up to 24 hours in the refrigerator.
Lightly dust a work surface with semolina flour. Take the dough ball and roll it out to a 1-inch thickness. Cut the dough into 1/2-inch-thick slices and roll out to a 1/4-inch thickness. Run the long strips through the cavatelli machine to shape. Depending on your type of machine, the thickness of the long strips may vary slightly. The pasta can then be refrigerated in an airtight container for up to 2 days or cooked immediately.
Boil a large pot of water over high heat. Add the remaining tablespoon salt and 6 cups of the pasta. Cook until slightly chewy, approximately 5 minutes. Drain the pasta, reserving 1 cup of the cooking liquid.
For the sauce verde: Place olive oil, spinach, capers, lemon juice, pepper and parsley in a tall, slender container, such as a pitcher, and blend on high speed using an immersion blender until the sauce has the consistency of a thick paste, about 1 minute.
To prepare the dish for plating, reheat the braised pork over medium heat. When the pork is steaming hot, add the cooked pasta. Allow the pork and pasta to cook another 2 to 3 minutes. If the sauce thickens too much, add some of the reserved pasta cooking water 1/4 cup at a time until you reach your desired consistency. Evenly divide the pork and pasta among 6 to 8 bowls. Spoon 1 tablespoon sauce verde over each bowl and top with a liberal grating of cheese.
The pork, pasta and sauce verde can be prepared up to 48 hours in advance and then finished when you are ready to serve.
The remaining sauce verde can be refrigerated for up to 7 days. It's a delicious condiment with grilled or roasted meats and vegetables, or simply spooned over scrambled eggs.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.